An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB= C

Abstract

An iteration method is constructed to solve the linear matrix equation AXB= C over symmetric X. By this iteration method, the solvability of the equation AXB= C over symmetric X can be determined automatically, when the equation AXB= C is consistent over symmetric X, its solution can be obtained within finite iteration steps, and its least-norm symmetric solution can be obtained by choosing a special kind of initial iteration matrix, furthermore, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new matrix equation A XB= C . Finally, numerical examples are given for finding the symmetric solution and the optimal approximation symmetric solution of the matrix equation AXB= C.